Implications

This non-implication, Form 0 \( \not \Rightarrow \) Form 31, whose code is 4, is constructed around a proven non-implication as follows:

  • This non-implication was constructed without the use of this first code 2/1 implication.
  • A proven non-implication whose code is 3. In this case, it's Code 3: 199, Form 0 \( \not \Rightarrow \) Form 209 whose summary information is:
    Hypothesis Statement
    Form 0  \(0 = 0\).

    Conclusion Statement
    Form 209 <p> There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 10166, whose string of implications is:
    31 \(\Rightarrow\) 209

The conclusion Form 0 \( \not \Rightarrow \) Form 31 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal M28\) Morris' Model II Morris constructs a generic extension of acountable standard model of ZFC in which there is a proper class ofgeneric sets

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